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Assume thatbe a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent  P . In this paper, it is shown that under certain conditions,wherean additive map onand  AB  =  P  if and only if it has the form, where  h  is an additive map into its center vanishing at commutatorswithand. An application of our results we characterize generalized Lie derivations on. Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras.

Lie Centralizers and generalized Lie derivations on prime rings by local actions